Here is the problem: The time it takes for a baker to bake a loaf of bread without it being underbaked is normally distributed with a mean of µ minutes and a standard deviation of σ minutes. Bread is considered to be slightly baked if it is baked for longer than (µ + 1.5σ) minutes. What is the probability that bread randomly selected by the baker will be slightly toasted?
Possibile answers: A. 93,32% B. 69,12% C. 30,85% D. 6,68%
I wonder od if it is possibile(and how)to calculate it so I don't have to look at normal distribution curve. I tried to look for some information on the internet but its didn't help.
I think the empirical rule can be applied here.
Remember that in normal distributions (with a random variable $T$ - time to bake a loaf of bread without underbaking it) $$P(\mu-\sigma<T<\mu+\sigma)\approx0.68$$ $$P(\mu-2\sigma<T<\mu+2\sigma)\approx0.95$$ Now, if you imagine the normal distribution curve in your mind, it is symmetric over the mean $\mu$, so the probabilities of time being outside these ranges (e.g. below $\mu-\sigma$ or above $\mu+\sigma$) are just $1-P(\text{inside these ranges})$, and (because these probabilities are equal due to symmetry) we divide by $2$ to get the probability of time being bigger than the upper bound of these ranges $$P(T>\mu+\sigma)\approx\frac{1-0.68}{2}\approx0.16\quad(16\%)$$ $$P(T>\mu+2\sigma)\approx\frac{1-0.95}{2}\approx0.025\quad(2.5\%)$$ So, the probability of $T$ being greater than $\mu+1.5\sigma$ has to be between these two percentages. The only answer that applies is D - $6.68\%$.
Hope this helps!